How to transition from plane to cylinder
Steps:


Enable LoopTools add-on.
Create Circle with chosen vertices count. Mine is 16.
Go to Edit Mode rotate Circle 90 degrees (X or Y axis).
Select top most vertex on the Circle and hit V to Rip selected vertex to two vertices. Now Circle has 17 vertices.
Create line with 17 vertices. I've done it by adding a Plane, deleted two vertices and CtrlR > 15 (15 cuts + 2 verts = 17).
Level line with Circle bottom. I've used snapping. And also scale it a bit. 





Select All A than W > LoopTools > Bridge and set number of Segments and Strength of this bridge.
Extrude line to have a plane and extrude circle to have a tube.